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For any primitive $n$-th root $\\zeta$ of unity, we obtain closed formulas for the permanents $$\\mathrm{per}\\left[1-\\zeta^jx_k\\right]_{1\\le j,k\\le n}\\ \\ \\text{and}\\ \\ \\mathrm{per}\\left[\\frac1{1-\\zeta^{j-k}x}\\right]_{1\\le j,k\\le n}.$$ Another typical result states that for any odd integer $n>1$ we have $$t_n:=\\frac1{\\sqrt n}\\mathrm{per}\\left[\\tan\\pi\\frac{jk}n\\right]_{1\\le j,k\\le (n-1)/2}\\in\\mathbb Z,$$ and that $t_p\\equiv(-1)^{(p+1)/2}\\pmod p$ for any odd prime $p$. We also pose severa"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2108.07723","kind":"arxiv","version":7},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2021-08-17T16:59:16Z","cross_cats_sorted":[],"title_canon_sha256":"5fa976ca1ab86fa473ad6e1470bd96e1dbfdecc42e4ef77fefa234b2c9ae3d14","abstract_canon_sha256":"1a413c8a435f957ee429e77a94e4fc6c6de3c864a5db53b5fbaaabe7f84e28ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:29:08.900892Z","signature_b64":"1p1UsG273B2uQ9uv33/lcwnX6Id3L3qetgpw4d226iJMEUpdzYUDBFF6iogCrjqxbJ+l2ZG+F0u7kFiWMQJnBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"629d75bce4bcbf7aa0ed6a6ea7a21b0b36e45acf5777a61fd4018c9e04950f2c","last_reissued_at":"2026-07-05T04:29:08.900408Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:29:08.900408Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arithmetic properties of some permanents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Zhi-Wei Sun","submitted_at":"2021-08-17T16:59:16Z","abstract_excerpt":"In this paper we study arithmetic properties of some permanents, many of which involve trigonometric functions. For any primitive $n$-th root $\\zeta$ of unity, we obtain closed formulas for the permanents $$\\mathrm{per}\\left[1-\\zeta^jx_k\\right]_{1\\le j,k\\le n}\\ \\ \\text{and}\\ \\ \\mathrm{per}\\left[\\frac1{1-\\zeta^{j-k}x}\\right]_{1\\le j,k\\le n}.$$ Another typical result states that for any odd integer $n>1$ we have $$t_n:=\\frac1{\\sqrt n}\\mathrm{per}\\left[\\tan\\pi\\frac{jk}n\\right]_{1\\le j,k\\le (n-1)/2}\\in\\mathbb Z,$$ and that $t_p\\equiv(-1)^{(p+1)/2}\\pmod p$ for any odd prime $p$. We also pose severa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.07723","kind":"arxiv","version":7},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2108.07723/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2108.07723","created_at":"2026-07-05T04:29:08.900467+00:00"},{"alias_kind":"arxiv_version","alias_value":"2108.07723v7","created_at":"2026-07-05T04:29:08.900467+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.07723","created_at":"2026-07-05T04:29:08.900467+00:00"},{"alias_kind":"pith_short_12","alias_value":"MKOXLPHEXS7X","created_at":"2026-07-05T04:29:08.900467+00:00"},{"alias_kind":"pith_short_16","alias_value":"MKOXLPHEXS7XVIHN","created_at":"2026-07-05T04:29:08.900467+00:00"},{"alias_kind":"pith_short_8","alias_value":"MKOXLPHE","created_at":"2026-07-05T04:29:08.900467+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.03970","citing_title":"Some new results on determinants and permanents","ref_index":6,"is_internal_anchor":false},{"citing_arxiv_id":"2605.16240","citing_title":"Evaluation of two determinants involving $q$-integers","ref_index":8,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MKOXLPHEXS7XVIHNNJXKPIQ3BM","json":"https://pith.science/pith/MKOXLPHEXS7XVIHNNJXKPIQ3BM.json","graph_json":"https://pith.science/api/pith-number/MKOXLPHEXS7XVIHNNJXKPIQ3BM/graph.json","events_json":"https://pith.science/api/pith-number/MKOXLPHEXS7XVIHNNJXKPIQ3BM/events.json","paper":"https://pith.science/paper/MKOXLPHE"},"agent_actions":{"view_html":"https://pith.science/pith/MKOXLPHEXS7XVIHNNJXKPIQ3BM","download_json":"https://pith.science/pith/MKOXLPHEXS7XVIHNNJXKPIQ3BM.json","view_paper":"https://pith.science/paper/MKOXLPHE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2108.07723&json=true","fetch_graph":"https://pith.science/api/pith-number/MKOXLPHEXS7XVIHNNJXKPIQ3BM/graph.json","fetch_events":"https://pith.science/api/pith-number/MKOXLPHEXS7XVIHNNJXKPIQ3BM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MKOXLPHEXS7XVIHNNJXKPIQ3BM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MKOXLPHEXS7XVIHNNJXKPIQ3BM/action/storage_attestation","attest_author":"https://pith.science/pith/MKOXLPHEXS7XVIHNNJXKPIQ3BM/action/author_attestation","sign_citation":"https://pith.science/pith/MKOXLPHEXS7XVIHNNJXKPIQ3BM/action/citation_signature","submit_replication":"https://pith.science/pith/MKOXLPHEXS7XVIHNNJXKPIQ3BM/action/replication_record"}},"created_at":"2026-07-05T04:29:08.900467+00:00","updated_at":"2026-07-05T04:29:08.900467+00:00"}